90 N. H. Bingham <strong>and</strong> A. J. OstaszewskiIn this case, as asserted,A ∩ ψ −1u (A) ≠ ∅.In the other case (w ∈ W ′ \M ′ ), one obtains similarlyHere tooψ u (w) ∈ V \M ⊆ A.A ∩ ψ −1u (A) ≠ ∅.Remarks. 1. In the theorem above it is possible to work with a weaker condition, namelylocal convergence at z 0 , where one dem<strong>and</strong>s that for some neighbourhood B η (z 0 ) <strong>and</strong>some K,d(ψ u (z), z) ≤ Kd(u, u 0 ), for z ∈ B η (z 0 ).This implies that, for any ε > 0, there is δ > 0 such that, for z ∈ B δ (z 0 ),d(ψ u (z), z) < ε, for z ∈ B δ (z 0 ).2. The Piccard-Pettis Theorem for <strong>topological</strong> <strong>groups</strong> (named by Kelley, [Kel, Ch. 6Pblm P-(b)], the Banach-Kuratowski-Pettis Theorem, say BKPT for short) asserts thecategory version of the Steinhaus Theorem [St] that, for A Baire <strong>and</strong> non-meagre, the setA −1 A is a neighbourhood of the identity; our version of the Piccard theorem as statedimplies this albeit only in the context of metric <strong>groups</strong>. Let d X be a right-invariantmetric on X <strong>and</strong> take ψ u (x) = ux <strong>and</strong> u 0 = e. Then ψ u converges to the identity (since‖ψ u ‖ := sup x d(ux, x) = d(u, e) = ‖u‖), <strong>and</strong> so the theorem implies that B δ (e) ⊆ A −1 Afor some δ > 0; indeed a ′ ∈ A ∩ ψ u (A) for u ∈ B δ (e) means that a ′ ∈ A <strong>and</strong>, for somea ∈ A, also ua = a ′ so that u = a −1 a ′ ∈ A −1 A. It is more correct to name the followingimportant <strong>and</strong> immediate corollary the BKPT, since it appears in this formulation in[Ban-G], [Kur-1], derived by different means, <strong>and</strong> was used by Pettis in [Pet1] to deducehis Steinhaus-type theorem.Theorem 6.15 (McShane’s Interior Points Theorem – [McSh, Cor. 3]). For X a <strong>topological</strong>space, let T : X 2 → X be such that T a (x) := T (x, a) is a self-homeomorphismfor each a ∈ X such that for each pair (x 0 , y 0 ) there is a self-homeomorphism ϕ withy 0 = ϕ(x 0 ) satisfyingT (x, ϕ(x)) = T (x 0 , y 0 ), for all x ∈ X.Let A <strong>and</strong> B be second category with B Baire. Then the image T (A, B) has interior points<strong>and</strong> there are A 0 ⊆ A, B 0 ⊆ B, with A\A 0 <strong>and</strong> B\B 0 meagre <strong>and</strong> T (A 0 , B 0 ) open.Remark. Despite it very general appearance, Th. 6.15 has little to offer in the normedgroup context. Indeed for a normed group X with right topology <strong>and</strong> with T (x, y) = xy −1each section T a is a homeomorphism, being the right-shift ρ a −1, which is a homeomorphism.However, the equation xy −1 = c has solution function y = ϕ(x) = c −1 x, a
<strong>Normed</strong> <strong>groups</strong> 91left-shift, not in general even continuous. The alternative T (x, y) = xy −1 xy −1 introducesshift operations to the left of the second x.7. The Kestelman-Borwein-Ditor Theorem: a bi<strong>topological</strong>approachIn this section we develop a bi-<strong>topological</strong> approach to a generalization of the KBDTheorem (Th. 1.1). An alternative approach is given in the next section. Let (X, S, m)be a probability space which is totally finite. Let m ∗ denote the outer measurem ∗ (E) := inf{m(F ) : E ⊂ F ∈ S}.Let the family {K n (x) : x ∈ X} ⊂ S satisfy (i) x ∈ K n (x), (ii) m(K n (x)) → 0.Relative to a fixed family {K n (x) : x ∈ X} define the upper <strong>and</strong> lower (outer) density atx of any set E byD ∗ (E, x) = sup lim sup n m ∗ (E ∩ K n (x))/m(K n (x)),D ∗ (E, x) = inf lim inf n m ∗ (E ∩ K n (x))/m(K n (x)).By definition D ∗ (E, x) ≥ D ∗ (E, x). When equality holds, one says that the densityof E exists at x, <strong>and</strong> the common value is denoted by D ∗ (E, x). If E is measurable thestar associated with the outer measure m ∗ is omitted. If the density is 1 at x, then xis a density point; if the density is 0 at x, then x is a dispersion point of E. Say that a(weak) density theorem holds for {K n (x) : x ∈ X} if for every set (every measurable set)A almost every point of A is a density (an outer density) point of A. Martin [Mar] showsthat the familyU = {U : D ∗ (X\U, x) = 0, for all x ∈ U}forms a topology, the density topology on X, with the following property.Theorem 7.1 (Density Topology Theorem). If a density theorem holds for {K n (x) : x ∈X} <strong>and</strong> U is d-open, then every point of U is a density point of U <strong>and</strong> so U is measurable.Furthermore, a measurable set such that each point is a density point is d-open.We note that the idea of a density topology was introduced slightly earlier by Goffman([GoWa],[GNN]); see also Tall [T]. It can be traced to the work of Denjoy [Den] in1915. Recall that a function is approximately continuous in the sense of Denjoy iff it iscontinuous under the density topology: [LMZ, p. 1].Theorem 7.2 (Category-Measure Theorem – [Mar, Th. 4.11]). Suppose X is a probabilityspace <strong>and</strong> a density theorem holds for {K n (x) : x ∈ X}. A necessary <strong>and</strong> sufficientcondition that a set be nowhere dense in the d-topology is that it have measure zero.Hence a necessary <strong>and</strong> sufficient condition that a set be meagre is that it have measurezero. In particular the <strong>topological</strong> space (X, U) is a Baire space.
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N. H. BINGHAM and A. J. OSTASZEWSKI
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Normed groups 3ContentsContents . .
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1. IntroductionGroup-norms, which b
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Normed groups 3Topological complete
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Normed groups 5abelian group has se
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Normed groups 74 (Topological permu
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Normed groups 9The following result
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Normed groups 11Corollary 2.4. For
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Normed groups 13More generally, for
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Normed groups 15definitions, our pr
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Normed groups 17so that fg is in th
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Normed groups 19(iii) The ¯d H -to
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Normed groups 21so‖αβ‖ ≤
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Normed groups 23Remark. Note that,
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Normed groups 25shows that [z n , y
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Normed groups 27Denoting this commo
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Normed groups 29Theorem 3.4 (Equiva
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Normed groups 31argument as again p
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Normed groups 33(ii) For α ∈ H u
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Normed groups 35Definition. A group
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Normed groups 37We now give an expl
- Page 43 and 44: Normed groups 39Theorem 3.19 (Abeli
- Page 45 and 46: Normed groups 412. Further recall t
- Page 47 and 48: Normed groups 43Theorem 3.22 (Lipsc
- Page 49 and 50: Normed groups 45Proof. Z γ = G (cf
- Page 51 and 52: Normed groups 47Theorem 3.30. Let G
- Page 53 and 54: Normed groups 49Remark. On the matt
- Page 55 and 56: Normed groups 51As for the conclusi
- Page 57 and 58: Normed groups 53By (C-adm), we may
- Page 59 and 60: Normed groups 55equipped with an in
- Page 61 and 62: Normed groups 57Proof. To apply Th.
- Page 63 and 64: Normed groups 59Definition. A point
- Page 65 and 66: Normed groups 61Proposition 3.46 (M
- Page 67 and 68: Normed groups 63Thus ω δ (s) ≤
- Page 69 and 70: Normed groups 65Remark. In the penu
- Page 71 and 72: Normed groups 67The result confirms
- Page 73 and 74: Normed groups 69Proof. By the Baire
- Page 75 and 76: Normed groups 715. Generic Dichotom
- Page 77 and 78: Normed groups 73Returning to the cr
- Page 79 and 80: Normed groups 75Examples. Here are
- Page 81 and 82: Normed groups 77cf. [Eng, 4.3.23].)
- Page 83 and 84: Normed groups 79Remarks. 1. See [Fo
- Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
- Page 87 and 88: Normed groups 83is continuous at th
- Page 89 and 90: Normed groups 85compact. Evidently,
- Page 91 and 92: Normed groups 87j ∈ ω} which enu
- Page 93: Normed groups 89The result below ge
- Page 97 and 98: Normed groups 93As a corollary of t
- Page 99 and 100: Normed groups 953. For X a normed g
- Page 101 and 102: Normed groups 97Proof. Note that‖
- Page 103 and 104: Normed groups 99Taking h(x) := ‖
- Page 105 and 106: Normed groups 1019. The Semigroup T
- Page 107 and 108: Normed groups 103Theorem 9.5 (Semig
- Page 109 and 110: Normed groups 105By the Category Em
- Page 111 and 112: Normed groups 107Proof. Say f is bo
- Page 113 and 114: Normed groups 109Thus G is locally
- Page 115 and 116: Normed groups 111Theorem 10.10 (Bar
- Page 117 and 118: Normed groups 113K-analyticity was
- Page 119 and 120: Normed groups 115Theorem 11.6 (Disc
- Page 121 and 122: Normed groups 117restricted to X\M
- Page 123 and 124: Normed groups 119groups need not be
- Page 125 and 126: Normed groups 121Proof. In the meas
- Page 127 and 128: Normed groups 123Hence, as t i n
- Page 129 and 130: Normed groups 125The corresponding
- Page 131 and 132: Normed groups 127(t, x) ✛✻Φ T
- Page 133 and 134: Normed groups 129Fix s. Since s is
- Page 135 and 136: Normed groups 131Hence,‖x‖ −
- Page 137 and 138: Normed groups 133converging to the
- Page 139 and 140: Normed groups 135Definition. Let {
- Page 141 and 142: Normed groups 137However, whilst th
- Page 143 and 144: Normed groups 139embeddable, 14enab
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Normed groups 141Bibliography[AL]J.
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Normed groups 143Series 378, 2010.[
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Normed groups 145abelian groups, Ma
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Normed groups 147[Kak] S. Kakutani,
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Normed groups 149fields. I. Basic p
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Normed groups 151[So]R. M. Solovay,